The horse could have nothing to do with the Garoghlanian family because they have a reputation for honesty.

When Mourad shows up with the white horse, Aram is stunned. He believes that the horse couldn't have anything to do with his cousin, himself, or his other family members. This is because the reputation of their family is one of poverty but honesty. He says the family is first proud, then honest, then believes in right and wrong. It's what they're known for. No one would ever think that any of the Garoghlanians would steal a horse.

When the horse's owner sees the horse with the Garoghlanian family, he believes in their honest reputation so much that he accepts that it's a different horse. He says that he looks with his heart instead of his eyes.

While the beautiful white horse showed up mysteriously, there was never an assumption that anyone had stolen it or done anything nefarious to acquire it. The Garoghlanian family is extremely poor, but it is still never assumed that they had done anything unlawful to get the horse with them. They are well known for their honesty and trust, and because of this, the whole region has confidence in them and would trust them if they said they did not take the horse. Additionally, the parents of the family knew nothing of the horse, it was the boys who brought it to their attention, and so people rightfully assume that the horse was not stolen and had simply wandered off. It was a fortunate accident that the horse ended up on their property.